Theorem REF  asserts that the proposed Byzantine-robust decentralized stochastic optimization method can reach a neighborhood of the optimal solution \(x^*\)  of (REF ). The convergence rate is sublinear and matches the rates of Byzantine-free decentralized stochastic optimization methods [1]}, [2]}, [3]}, [4]}. The size of neighborhood is proportional to \(p\)  (the dimension of model), \(\lambda ^2\)  (squared penalty parameter), and \(\sum _{i \in \mathcal {R}} |\mathcal {B}_i|^2\)  that is determined by the number of Byzantine agents and the network topology. Combining Theorems REF  and REF , we derive the main Theorem as follows.
